Instituto de Matem atica Pura e Aplicada - impa.br · Instituto de Matem atica Pura e Aplicada Vanessa Ribeiro Ramos EQUILIBRIUM STATES FOR HYPERBOLIC POTENTIALS Thesis presented

Instituto de Matematica Pura e Aplicada

Doctoral Thesis

EQUILIBRIUM STATES FOR HYPERBOLIC POTENTIALS

Vanessa Ribeiro Ramos

Rio de Janeiro

May 22, 2013

Instituto de Matematica Pura e Aplicada

Vanessa Ribeiro Ramos

EQUILIBRIUM STATES FOR HYPERBOLIC POTENTIALS

Thesis presented to the Post-graduate Program in Math-

ematics at Instituto de Matematica Pura e Aplicada as

partial fulfillment of the requirements for the degree of

Doctor in Philosophy in Mathematics.

Advisor: Marcelo Miranda Viana da Silva

Rio de Janeiro

2013

To God.

Acknowledgments

Agradeco a Deus, forca criadora que impulsiona e conduz a minha vida.

Aos meus amados pais, Avani e Edvaldo, agradeco pela educacao moral e

pelo amor incondicional. Por este amor, eles abdicaram da minha presenca

em funcao da minha felicidade.

Ao meu amado, Ivaldo Nunes, obrigada por estar sempre ao meu lado. Com

muito amor e cumplicidade, estamos construindo uma linda vida juntos!

Agradeco ao meu orientador, Prof. Marcelo Viana, pelo apoio e orientacao

deste trabalho. O convıvio ao longo destes anos mostrou que alem de exce-

lente matematico, ele e admiravel pela gentileza e hombridade.

Ao professor Enrique Pujals, obrigada pelo incentivo. A sua constante pre-

senca e entusiasmo transmitem confianca aos jovens dinamicistas.

Aos membros da banca, Prof. Carlos Gustavo Moreira, Prof. Enrique Pu-

jals, Prof. Javier Solano, Prof. Marcelo Viana, Prof. Nivaldo Muniz e Prof.

Vilton Pinheiro agradeco pelos enriquecedores comentarios e sugestoes ao

trabalho.

Aos meus queridos “baianos”, Prof. Armando Castro, Prof. Paulo Varandas,

Prof. Vilton Pinheiro, Prof. Vitor Araujo, o meu profundo agradecimento.

Tive a felicidade de conhece-los ainda na UFBA e desde entao, compartilho

da amizade e do incentivo deles. Certamente, eles sao os grandes responsaveis

por eu ter chegado ao IMPA.

Agradeco tambem aos funcionarios do IMPA. O crescimento e fortalecimento

desta instituicao e consequencia da atenciosa e coesa equipe que eles formam.

Um agradecimento muito especial as minhas amigas Daphne e Maria Teresa

iv

Vanessa Ribeiro Ramos Equilibrium States for Hyperbolic Potentials

Gilly. A acolhida e amizade delas fizeram a minha estadia no Rio de Janeiro

muito mais feliz e tranquila.

Nao poderia deixar de agradecer a amizade e o carinho das minhas queri-

das amigas que me acompanham desde a graduacao, as “super-poderosas”

Eliane, Fabiana, Manuela e Isis. Nossa amizade e eterna!

Agradeco aos meus estimados amigos do IMPA e alem IMPA, pelos conselhos

e ensinamentos durante estes anos. A presenca de cada um deles confortou e

alegrou momentos tensos e solitarios. Mesmo correndo o risco de esquecer o

nome de alguma pessoa que e importante para mim, quero destacar os ami-

gos: Alan Prata, Almir Rogerio, Ana Maria Menezes, Areli Vazquez, Artem

Raibekas, Edilaine Nobili, Elaıs Cidely, Felipe Nobili, Jaqueline Siqueira,

Jose Manuel, Jose Regis, Lucas Ambrozio, Lucas Backes, Luciana Salgado,

Maria Andrade, Mariana Pinheiro, Mario Roldan, Pablo Guarino, Pedro

Hernandez, Priscilla Fustinoni, Rafael Montezuma, Renan de Lima, Ricardo

Turolla, Roseane Alves, Samuel Barbosa, Sergio Ibarra, Tiane Marcarini,

Vanessa Simoes, Vinicius Albani, Wanderson Costa, Yuri Lima e Yuri Ki.

Ao CNPq, agradeco pelo apoio financeiro.

Enfim, muito obrigada a todos que contribuıram para o exito desta etapa em

minha vida.

Instituto de Matematica Pura e Aplicada v May 22, 2013

Abstract

The work deals with the problem of existence and uniqueness of equilibrium

states associated to local diffeomorphisms. We obtain uniqueness of such

measures for hyperbolic potentials, which have topological pressure located

on a non-uniformly expanding region. We also show that partially hyper-

bolic local diffeomorphisms of type Ec⊕Eu with central dimension less than

or equal two dim(Ec) ≤ 2, dominated splitting Ec = Ec1 ⊕ Ec

2 into one-

dimensional subbundles and invariants central curves are entropy-expansive

and therefore, they have equilibrium states for any continuous potential.

Keywords: Equilibrium States, Local Diffeomorphisms, Hyperbolic

Potentials.

vi

Resumo

O trabalho trata o problema de existencia e unicidade de estados de equilıbrio

associados a difeomorfismos locais. Provamos a unicidade de tais medidas

para potenciais hiperbolicos, os quais possuem pressao topologica localizada

em uma regiao nao-uniformemente expansora. Mostramos tambem que difeo-

morfismos locais parcialmente hiperbolicos do tipo Ec⊕Eu com dimensao da

direcao central menor ou igual a dois dim(Ec) ≤ 2, decomposicao dominada

Ec = Ec1 ⊕ Ec

2 em subfibrados unidimensionais e curvas centrais invariantes

sao entropy-expansive e portanto, admitem estados de equilıbrio para qual-

quer potencial contınuo.

Palavras-chave: Estados de Equilıbrio, Difeomorfismos Locais, Potenciais

Hiperbolicos.

vii

Contents

Introduction 1

1 Preliminaries 5

1.1 Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Hyperbolic Potentials . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Natural Extension . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Existence of Equilibrium States 11

3 Uniqueness of Equilibrium States 19

3.1 Conformal Measures . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . 27

4 Partially Hyperbolic Skew-Products 32

Bibliography 39

Introduction

The theory of equilibrium states, developed by Sinai, Ruelle and Bowen in

the seventies and eighties, came into existence through the application of

techniques and results from statistical mechanics to smooth dynamics.

In the classical setting, given a continuous map f : M →M on a compact

metric space M and a continuous potential φ : M → R, we say that µφ is

an equilibrium state associated to (f, φ), if µφ is an f -invariant probability

measure characterized by the following variational principle:

Pf (φ) = hµφ(f) +

∫φdµφ = sup

µ∈Mf (M)

{hµ(f) +

∫φ dµ

}where Pf (φ) denotes the topological pressure, hµ(f) denotes the metric en-

tropy and the supremum is taken over all f -invariant probabilities measures.

In the uniformly hyperbolic context, which includes uniformly expanding

maps, equilibrium states always exist and they are unique if the potential φ

is Holder continuous and the dynamics f is transitive. However, the scenario

beyond hyperbolic systems is pretty much incomplete.

Recently, several advances were obtained outside the uniformly hyperbolic

setting in the works of Bruin and Keller [5] and Denker and Urbanski [8],

for interval maps and rational functions on the Riemann sphere; Leplaideur,

Oliveira, Rios [15] on partially hyperbolic horsehoes; and Buzzi and Sarig [7]

and Yuri [21], for countable Markov shifts and for piecewise expanding maps

in one and higher dimensions, among others.

1


Before proceeding a few words are in order on the nature of the two pro-

blems: existence and uniqueness of equilibrium states. Existence is relatively

soft property that can often be established via compacteness arguments.

Uniqueness is usually more subtle, and requires a better understanding of

the dynamics. Examples of transitive shifts with equilibrium states having

nontrivial supports go back to Krieger [13]. So, some conditions on the

potential is certainly necessary.

For local diffeomorphisms which present expansion in a non-uniform way

we point out the results of Oliveira and Viana [17]; Arbieto, Matheus and

Oliveira [2]; Varandas and Viana [20]. They obtained uniqueness of equili-

brium states for potentials with small oscillation. In this context, they were

able to construct expanding equilibrium states absolutely continuous with

respect to the conformal measure; moreover, the equilibrium state is unique.

The hypothesis on the potential is used to ensure that most of the pres-

sure emanates from regions of the ambient where the dynamics is actually

hyperbolic.

These findings motivate, to some extent, the approach that we follow in

the present paper. Indeed, we consider a certain inequality between the pres-

sure of (f, φ) on the hyperbolic and the nonhyperbolic regions of M , and we

prove that, in suitable situations, this is suffices for existence and uniqueness

of equilibrium states. The results mentioned in the previous paragraph fit in

this framework. Lets us state our results.

Let f : M →M be a C1 local diffeomorphism on a compact manifold M .

Given a positive constant σ, we denote by Σ = Σ(σ) the set whose points

satisfy:

Σ :=

{x ∈M ; lim sup

n→+∞

1

n

n−1∑i=0

log∥∥Df(f i(x))−1

∥∥ ≤ log σ

}

We say that a real continuous function φ : M → R is a hyperbolic potential

if there exists some σ ∈ (0, 1) such that

Pf (φ,Σc) < Pf (φ,Σ) = Pf (φ).

In this context, our main result is

Instituto de Matematica Pura e Aplicada 2 May 22, 2013


Theorem A. Let f : M → M be a transitive C1 local diffeomorphism on a

compact manifold M and let φ : M → R be a hyperbolic Holder continuous

potential. Then, there exists a unique equilibrium state µφ for (f, φ).

Hyperbolicity of the potential is the essential condition in our main result.

Closely related condition have been considered by Hofbauer and Keller [12],

Denker, Keller and Urbanski [9] for piecewise monotonic maps and by Buzzi,

Paccaut, Schmitt [6], in the context of piecewise expanding multidimensional

maps.

We also prove uniqueness of equilibrium states associated to skew-products

F (x, y) = (f(x), gx(y)) whose bases dynamics f : M →M is a transitive C1

local diffeomorphism and the fiber dynamics g : M × N → N is uniformly

contracting, and potentials φ which are hyperbolic on the bases M , that

means, PF (φ,Σc ×N) < PF (φ,Σ×N) = Pf (φ).

Theorem B. Let Λ be a partially hyperbolic attractor for the skew-product

F : M × N → M × N , F (x, y) = (f(x), gx(y)) and let φ : M × N → Rbe a Holder continuous potential hyperbolic on M . Then, there is only one

equilibrium state µφ associated to (F, φ).

Observe that the hyperbolicity condition of the potential φ imply positive

Lyapunov exponents for regular points x ∈ Σ. Therefore, since we have a

fiber contraction on N it makes sense only to require hyperbolicity condition

on M .

An existence theorem

As we pointed out before, existence can often be obtained from compactness.

One way to do this is through the notion of entropy-expansiveness. We say

that the dynamics f is entropy-expansive if there exists some ε > 0 such that

the topological entropy htop(B∞ε (x)) of the set whose points stay ε-next to

x, namely infinity ball B∞ε (x) := {y ; d(f j(x), f j(y)) ≤ ε for all j ∈ N}, is

equal to zero in every point.

In [4] Bowen proved that entropy-expansiveness implies upper-semiconti-

nuity of the pressure function and therefore, that kind of dynamics has equi-

librium measures. This condition is satisfied for hyperbolic dynamics, see



[3]; for diffeomorphisms far from homoclinic tangencies, see Liao, Viana, and

Yang [16]; for diffeomorphisms partially hyperbolic with one-dimensional cen-

tral subbundles, see Dıaz, Fisher, Pacıfico and Vieitez [10]. Roughly speak-

ing, in these situations thanks to the existence of (almost) invariant folia-

tions it is possible to localize the infinity ball B∞ε (x) on central leaves W c(x).

Hence, they conclude entropy-expansiveness from the sub-exponential growth

of the generators of B∞ε (x). However, in the absence of these foliations, for

example local diffeomorphisms, important difficulties arise. In this work, we

show that local diffeomorphisms partially hyperbolic are entropy-expansive.

More precisaly, we have

Theorem C. Let f : M →M be a partially hyperbolic local diffeomorphism

of type Ec⊕Eu with dim(Ec) ≤ 2. If the central direction Ec has a dominated

splitting Ec1⊕Ec

2 into one-dimensional subbundles and every central curve is

invariant then f is entropy-expansive.

Theorem C gives us existence of equilibrium states associated to (f, φ)

for any continuous potential φ. Of course, we can not expect uniqueness

in this generality. On the other hand, there exist several interesting classes

of local diffeomorphisms to which this theorem applies, for instance, local

diffeomorphisms derived from expanding maps by isotopy, diffeomorphisms

partially hyperbolics, perturbations of skew-products and others.

This work is organized as follows. In Chapter 1, we introduce some pre-

liminaries facts and notations which will be used throughout this work. In

Chapter 2 we prove Theorem C. In Chapter 3, we construct conformal mea-

sures and we prove Theorem A. Finally, in Chapter 4 we prove Theorem B.


CHAPTER 1

Preliminaries

The aim of this chapter is fix notations, give definitions and state some facts

which will be used throughout this work. This content may be omitted in a

first reading and the reader can return here whenever necessary.

1.1 Equilibrium States

Let f : M → M be a continuous transformation defined on a compact

metric space M. Let φ : M → R be a real continuous function that we call

by potential.

Given an open cover α for M we define the pressure Pf (φ, α) of φ with

respect to α by

Pf (φ, α) := limn→+∞

1

nlog inf

U⊂αn{∑U∈U

eφn(U)}

where the infimum is taken over all subcover U of αn =∨n≥0 f

nα and φn(U)

indicates supx∈U

n−1∑j=0

φ ◦ f j(x).

5


Definition 1. The topological pressure Pf (φ) of the potential φ with respect

to the dynamics f is defined by

Pf (φ) := limδ→0

{sup|α|≤δ

Pf (φ, α)}

where |α| denotes the diamenter of the open cover α.

An alternatively way to define topological pressure is through the notion

of dynamical balls. This approach is from dimension theory and it is very

useful to calculate the topological pressure of non-compact sets.

Fix δ > 0. Given n ∈ N and x ∈M let Bδ(x, n) the dynamical ball:

Bδ(x, n) := {y ∈M / d(f j(x), f j(y)) ≤ δ , for 0 ≤ j ≤ n}

Put S = M × N. Denote by FN the collection of dynamical balls

FN = {Bδ(x, n) / x ∈M and n ≥ N}

Given a f -invariant subset Λ of M , non necessarily compact, let U be a

finite or countable family of FN which cover Λ.

For every γ ∈ R let

mf (φ,Λ, δ, N, γ) = infU⊂FN

{∑

Bδ(x,n)∈U

e−γn+Snφ(Bδ(x,n))}

As N goes to infinity we define

mf (φ,Λ, δ, γ) = limN→+∞

mf (φ,Λ, δ, N, γ)

Taking the infimum over γ we obtain

Pf (φ,Λ, δ) = inf {γ / mf (φ,Λ, δ, γ) = 0}

The relative pressure Pf (φ,Λ) of a subset Λ of M is given by

Pf (φ,Λ) = limδ→0

Pf (φ,Λ, δ)

We refer the reader to [18] for more details and properties of the relative

pressure.

The relationship between the topological pressure Pf (φ) and the metric

entropy hµ(f) of the dynamics is given by



Theorem 1 (Variational Principle). Let Mf (M) be the set of probability

measures invariants by a continuous transformation f : M → M defined on

a compact metric space M and let φ : M → R be a continuous function.

Then,

Pf (φ) = supµ∈Mf (M)

{hµ(f) +

∫φ dµ

}The variational principle gives a natural way of choosing important mea-

sures of Mf (M).

Definition 2. A measure µφ of Mf (M) is called an equilibrium state for

(f, φ) if µφ is characterized by the variational principle:

Pf (φ) = hµφ(f) +

∫φ dµφ.

The existence problem of equilibrium states is often be established via

compactness arguments. One way to do this is through the notion of entropy-

expansiveness.

We say that the dynamics f is entropy-expansive if there exists some

ε > 0 such that the topological entropy htop(B∞ε (x)) of the set whose points

stay ε-next to x, namely infinity ball

B∞ε (x) := {y ; d(f j(x), f j(y)) ≤ ε for all j ∈ N}

is equal to zero in every point x ∈ M . In [4] Bowen proved that entropy-

expansiveness implies upper-semicontinuity of the pressure function and the-

refore, that kind of dynamics has equilibrium measures.

1.2 Hyperbolic Potentials

Let M be a compact manifold and f : M →M be a C1 local diffeomorphism.

Given a positive constant σ, we denote by Σ = Σ(σ) the set whose points

satisfy:

Σ :=

{x ∈M ; lim sup

n→+∞

1

n

n−1∑i=0


∥∥ ≤ log σ

}Note that, for σ ∈ (0, 1) we have f−1 is locally a contraction. In this way,

Σ is a non-uniformly expanding region on M . A probability measure µ (not

necessarily invariant) is expanding if µ(Σ) = 1.



Definition 3. We say that a real continuous function φ : M → R is a

hyperbolic potential if there exists σ ∈ (0, 1) such that

Pf (φ,Σc) < Pf (φ,Σ) = Pf (φ).

The above inequality allows us to explore the non-uniform expansion

property on Σ. In this direction, the key ingredient are the so called hyper-

bolic times.

Definition 4 (Hyperbolic Times). Given σ ∈ (0, 1) we say that n is a hy-

perbolic time for x if for each 1 ≤ k ≤ n− 1 we have

n−1∏j=n−k

‖Df(f j(x))−1‖ ≤ σj

The next lemma shows that every point in Σ has infinitely many hyper-

bolic times. In fact, these times appear with a positive frequence θ > 0. For

details see [1].

Lemma 1. Let f : M → M be a C1 local diffeomorphism of a compact

manifold M . If x ∈M satisfy

lim supn→+∞

1

n

n−1∑j=1

log ‖Df(f j(x))−1‖ ≤ log σ < 0.

Then, there exists θ > 0, that depends only on f and σ, and a sequence of

hyperbolic times 1 ≤ n1(x) ≤ · · · ≤ nl(x) ≤ n for x with l ≥ θn.

Moreover, if η is a probability measure for which the limit above holds in

almost everywhere then for every borelean A ⊂ M with positive η-measure

we have

lim infn→∞

1

n

n−1∑j=1

η(A ∩ Σj)

η(A)≥ θ

2

where Σj denotes the set whose points have j as hyperbolic time.

The next result point out that on hyperbolic times the dynamics f locally

behaves as if it were an expanding map: we see the existence of inverse

branches with uniform backward contraction on some small neighborhoods.



Lemma 2 (Distortion Control). There exists δ > 0 such that for every

n = n(x) hyperbolic time for x, the dynamical ball Bδ(x, n) is mapped diffeo-

morphically by fn onto the ball B(fn(x), δ) with

d(fn−j(y), fn−j(z)) ≤ σjd(fn(y), fn(z)) (1.1)

for every 1 ≤ j ≤ n and every y, z ∈ Bδ(x, n).

Moreover, given a Holder continuous potential ψ there exists a constant

K > 0 such that

K−1 ≤ e−Snψ(y)+Snψ(z) ≤ K (1.2)

for every y, z ∈ Bδ(x, n).

Note that, since the potential ψ is Holder continuous, the inequality (1.2)

above is a direct consequence of the local contraction until hyperbolic time n

given by the first property (1.1) of the Lemma. For a proof of the property

(1.1) see [1].

1.3 Natural Extension

Let (M,d) be a compact metric space and let f : M → M be a continuous

non-invertible transformation. Define the space

M = {x := (. . . , x2, x1, x0) ∈MN ; f(xi+1) = xi for all i ≥ 0}.

Fixed δ ∈ (0, 1) we can define the following metric on M :

dM(x, y) :=∑j≥0

δjdM(xj, yj).

The natural extension of f is the homeomorphism

f : M → M, f(x) = f(..., x2, x1, x0) = (..., x2, x1, x0, f(x0))

Let π : M →M be the natural projection, that mean,

π(x) = π(. . . , x2, x1, x0) = x0

Note that, π is continuous, surjective and (semi)conjugates f and f .



On ergodic point of view, given an ergodic measure µ defined on Borel

subsets of M there exists a unique measure µ defined on Borel subsets of M

such that π∗µ = µ, that mean,

µ(A) = µ(π−1(A)), for every measurable set A ⊂M.

Moreover, since

π−1(x) = {(. . . , x2, x1, x0) ; x0 = x}

we observe that htop(f, π−1(x)) = 0 for every x ∈ M because we can choose

a subset of π−1(x) with finite cardinality as n-generator for every n ∈ N.

Thus, we apply the Ledrappier-Walter’s formula, and we conclude that if

µ projects on µ then hµ(f) = hµ(f).

Theorem 2 (Ledrappier-Walter’s formula). Let M , M be compact metric

spaces and let f : M → M , f : M → M , π : M → M be continuous maps

such that π is surjective and π ◦ f = f ◦ π then

supµ;π∗µ=µ

hµ(f) = hµ(f) +

∫htop(f , π

−1(y)) dµ(y)

See [14] for a proof of this theorem and [19] for more details about natural

extension.


CHAPTER 2

Existence of Equilibrium States

We begin this work studying partially hyperbolic local diffeomorphisms of

type Ec ⊕ Eu with dim(Ec) ≤ 2. Since f is not a diffeomorphism it is

necessary to clarify the definition of partially hyperbolic.

Let f : M → M be a Cr, r ≥ 1, local diffeomorphism of a compact

manifold M . We denote by f−1 the inverse branches of f , that means, for

each open set U ⊂ M such that f : U → f(U) is a diffeomorphism onto its

image we have well defined f−1 : f(U) → U with f ◦ f−1(x) = x for every

x ∈ f(U).

A continuous cone field Cu = (Cux )x∈M defined on injectivity domains of

f is said to be unstable if Cu is strictly Df -invariant and for some positive

constant λ < 1 and every inverse branch we have:∥∥Df−1x vu

∥∥ ≤ λ‖vu‖ , for all vu ∈ Cux ;

We say that f is partially hyperbolic of type Ec⊕Eu if for some unstable

direction Eu ⊂ Cu there exist a continuous splitting of the tangent bundle

TM = Ec⊕Eu into non-trivial subspaces Ec, Eu such that for every inverse

branch and some choice of a Riemannian metric on M we have:

1. Eu is uniformly expanding:∥∥Df−1

∣∣Eu(x)

∥∥ ≤ λ;

2. Ec is dominated by Eu:∥∥Df ∣∣Ec(x)

∥∥ · ∥∥Df−1∣∣Eu(f(x))

∥∥ ≤ λ.

11


Owing to the domination property, the bundle Ec is unique and Df -

invariant. On the other hand, since f is non-invertible, in general, there is

no unique Df -invariant unstable direction.

We say that Ec decomposes into one dimensional subbundles with domi-

nated splitting if there are Df -invariants one-dimensional subspaces Eci such

that Ec = Ec1 ⊕ Ec

2 ⊕ · · · ⊕ Eck with∥∥∥Df ∣∣

Ec1⊕···⊕Eci (x)

∥∥∥ · ∥∥∥Df−1∣∣Eci+1⊕···⊕Eck⊕Eu(f(x))

∥∥∥ ≤ λ

for i ∈ {1, 2, ..., k}, any direction Eux ⊂ Cu

x fixed and every inverse branch.

Under the assumption of Ec decomposes into one-dimensional dominated

subbundles with invariants central curves we will show that these systems

are entropy-expansive and therefore they have equilibrium states for any

continuous potential. Our goal is to prove:

Theorem C. Let f : M →M be a partially hyperbolic local diffeomorphism

of type Ec⊕Eu with dim(Ec) ≤ 2. If the central direction Ec has a dominated

splitting Ec1⊕Ec

2 into one-dimensional subbundles and every central curve is

invariant then f is entropy-expansive.

Corollary. For f as in Theorem C and φ : M → R continuous potential

there is an equilibrium state associated to (f, φ).

From now dim(Ec) ≤ 2 and we are supposing that Ec has dominated

splitting Ec = Ec1 ⊕ Ec

2 and every central curve is invariant.

Let us start fixing some small δ0 > 0 such that for all x ∈M the neighbor-

hood B(x, δ0) of x is contained in a local chart and therefore we can identify

B(x, δ0) with an open set of Rn.

We will denote by γσloc(x) for σ ∈ {c1, c2, u} any solution in B(x, δ0) of

the ordinary differential equation given by{X ′(y) = Eσ(y), y ∈ B(x, δ0)

X(0) = x

where σ = ci means that we fix the field Eci for i ∈ {1, 2} and when σ = u

we are fixing some continuous field Eu in Cu.

Although there is no uniqueness, these solutions γσloc are C1 curves and

quasi-isometrics in the sense of the following lemma.



Lemma 3. There exist 0 < δ1 ≤ δ0 such that for all δ ≤ δ1 if γσloc(x) is

contained in B(x, δ) then dγσloc(x, y) ≤ 2d(x, y) for any point y ∈ γσloc(x).

Proof. Due to the continuity of Ec and Cu, for all θ small there exist δ1 > 0

satisfying ∠(Eσ(z), Eσ(w)) < θ provided d(z, w) ≤ δ1. Hence, any curve

γσloc(x) in B(x, δ) for δ ≤ δ1 is a graph of a C1 function ξ

γσloc : [0, 1]→M, γσloc(t) = (x1(t), ξ ◦ x1(t))

with ‖∇ξ‖ ≤ tan 2θ because γσloc is tangent to Eσ.

Fixing θ small enough, we obtain for any y = γσloc(t) :

dγσloc(x, y) =

∫ t

0

∥∥(γσloc)′(s)∥∥ds =

∫ t

0

√(x′1(s))2 + ((ξ ◦ x1)′(s))2ds

=

∫ x1(t)

x1(0)

√1 + ‖∇ξ‖2ds

≤∫ x1(t)

x1(0)

√1 + (tan 2θ)2ds

≤ 2 ‖x1(t)− x1(0)‖ ≤ 2d(x, y)

Note that from the uniform transversality of Ec and Cu we can fix some

δ2 ≤ δ1 such that

d(x, y) ≤ δ2 =⇒ d(x, yσ) ≤ k0d(x, y)

where k0 is a constant which depends only on the metric and yσ is a point

of intersection yσ = γcloc(y) ∩ γσloc(x) for any central curve γciloc(y) of y with

i ∈ {1, 2} and γσloc(x) for σ ∈ {c1, c2, u}.Hence from the Lemma 3 follows if y ∈ B∞ε (x) then any point yc of the

intersection γciloc(x)∩γcjloc(y), i, j ∈ {1, 2}, is a element of B∞k0ε(x). In fact, the

invariance of central curves imply for all n > 0 :

fn(yc) = fn (γciloc(x)) ∩ fn(γcjloc(y)

)= γciloc(f

n(x)) ∩ γcjloc(fn(y)) = (fn(y))c

thus, for ε ≤ δ2 we have for all n > 0 :

d(fn(yc), fn(x)) ≤ k0d(fn(y), fn(x)) ≤ k0ε.



Now, by uniform continuity of f and Df , we also fix ε0 < δ2/4 and

δ3 < ε0/2 such that d(z, w) ≤ δ3 =⇒ d(f(z), f(w)) < ε0 and

(1− ζ) ≤∥∥Df |Ec1(z)

∥∥∥∥Df |Ec1(w)

∥∥ ≤ (1 + ζ) and (1− ζ) ≤∥∥Df−1|Ec2⊕Eu(z)

∥∥∥∥Df−1|Ec⊕Eu(w)

∥∥ ≤ (1 + ζ)

where ζ > 0 satisfy (1 + ζ)√λ < 1.

Let ε1 � δ3/2 small enough. The next proposition shows that any point

y ∈ B∞ε (x) has no projection on unstable curves of x.

Proposition 1. For all ε < ε1/2k0 if y ∈ B∞ε (x) then γcloc(y)∩γuloc(x) ⊂ {x}for any central curve γcloc(y) of y and any unstable curve γuloc(x) of x.

Proof. By contradiction, suppose that y ∈ B∞ε (x) and there exist yu =

γcloc(y) ∩ γuloc(x) for some γcloc(y) and γuloc(x).

If yu = y by Lemma 3, for any z in the segment [x, y]γu of γuloc(x) we have

dγu(x, z) ≤ dγu(x, y) ≤ 2ε and by uniform continuity,

df(γu)(f(x), f(z)) ≤ df(γu)(f(x), f(y)) ≤ 2ε0

Since γuloc is tangent to an expanding cone Cu we may take the first iterate

f j of f such that 2ε < `(f j(γu)) ≤ 2ε0 and by Lemma 3 again we obtain the

contradiction d(f j(x), f j(y)) > ε.

If yu 6= y the inequality d(x, yu) ≤ k0d(x, y) ≤ k0ε < ε1/2 gives us

dγu(x, yu) ≤ ε1. So, we repeat the previous argument and find j such that

2ε1 ≤ dfj(γu)(fj(x), f j(y)) ≤ 2ε0 and we obtain the same contradiction

d(f j(x), f j(y)) ≥ 1

k0

d(f j(x), f j(y)) ≥ ε1

k0

> 2ε

Therefore we must have γcloc(y) ∩ γuloc(x) ⊂ {x}.

For the next proposition we are supposing that Ec is one-dimensional. We

will show that B∞ε (x) is essentially located in only one central curve γcloc(x).

Proposition 2. Given ε < ε1/6k0 if γcloc(x) and ψcloc(x) are central curves

of x such that γcloc(x) ∩B∞ε (x) 6= {x} and ψcloc(x) ∩B∞ε (x) 6= {x} then

ψcloc(x) ∩B∞ε (x) ⊂ γcloc(x) or γcloc(x) ∩B∞ε (x) ⊂ ψcloc(x)



Proof. Firstly we note that if y ∈ γcloc(x) ∩B∞ε (x) then the segment [x, y] in

γcloc(x) is contained in B∞2ε (x).

In fact, due to Df -invariance of Ec, the curve f j(γcloc) is a solution in

B(f j(x), ε) of ODE{X ′(f j(z)) = Df jEc(z) = Ec(f j(z)),

X(0) = f j(x)

Since d(f j(x), f j(y)) ≤ ε for all j > 0 we have for any z ∈ [x, y] :

d(f j(x), f j(z)) ≤ dfj(γc)(fj(x), f j(z)) ≤ dfj(γc)(f

j(x), f j(y)) ≤ 2ε

where in the last inequality we used the Lemma 3.

Now suppose that there are y and z such that,{y ∈ γcloc(x) ∩B∞ε (x) but y /∈ ψcloc(x)

z ∈ ψcloc(x) ∩B∞ε (x) but z /∈ γcloc(x)

Because γuloc(y) and ψcloc(x) are tangents to transversal fields and Ec is one-

dimensional, replacing y to z if necessary, there exist zu ∈ γuloc(y) ∩ ψcloc(x)

with zu ∈ [x, z] and zu 6= y for some γuloc(y). And, as we saw above zu ∈B∞2ε (x) ∩ γuloc(y).

In this way, d(f j(zu), fj(y)) ≤ 3ε ≤ ε1/k0 for all j > 0 imply that

zu ∈ B∞3ε (y) ∩ γuloc(y) which is a contradiction by the previous proposition.

The last lemma of this chapter is a general result, independent of dim(Ec),

which is a consequence of the domination property of dynamics. This lemma

will allows us to find generators for B∞ε (x) of cardinality subexponential.

Lemma 4. For every ε < ε1/6k20 there are constants λ ∈ (λ, 1) and n0 =

n0(x) ∈ N such that fixed some central curve γc2loc(x) of x if y ∈ B∞ε (x) then

any point yc ∈ γc2loc(x) ∩ γc1loc(y) satisfy

d(fn(y), fn(yc)) ≤ λnd(y, yc) for all n ≥ n0.

Proof. Given y ∈ B∞ε (x) let y = γc2loc(y) ∩ γc1loc(x). By Lemma 3, we known

that y ∈ B∞k0ε(x). We claim that for any λ ∈ (λ,√λ) there exist n0 > 0 such

thatn−1∏j=0

∥∥∥Df−1∣∣Ec2⊕Eu(fj(z))

∥∥∥ > λn



for all n ≥ n0 and z in the segment [x, y] of γc,1loc(x).

Indeed, if for some λ ∈ (λ,√λ) the inequality is false for any n ∈ N then

there exist a sequence {zn}n∈N ∈ [x, y] such that

∥∥∥Df−n∣∣Ec2⊕Eu(fn(zn))

∥∥∥ ≤ n−1∏j=0

∥∥∥Df−1∣∣Ec2⊕Eu(fj(zn))

∥∥∥ ≤ λn.

Thus,

dγc,1loc(x)(x, y) ≤ λndγc,1loc(fn(x))(fn(x), fn(y)) ≤ ((1 + ζ)λ)nk0ε

n→+∞−→ 0

because the curve f j([x, y]) stays k0ε next to f j(zn) for all j > 0 and by

continuity of the derivatve we have

∥∥∥Df−n∣∣Ec2⊕Eu(fn(z))

∥∥∥ ≤ n−1∏j=0

∥∥∥Df−1∣∣Ec2⊕Eu(fj(z))

∥∥∥≤

n−1∏j=0

(1 + ζ)∥∥∥Df−1

∣∣Ec2⊕Eu(fj(zn))

∥∥∥ ≤ ((1 + ζ)λ)n

for all z in [x, y] and this prove our claim.

Now, fix some λ ∈ (λ,√λ) such that λ = λλ−1(1 + ζ) < 1. Apllying the

domination property

n−1∏j=0

∥∥∥Df j∣∣Ec1(·)

∥∥∥ · ∥∥∥Df−j∣∣Ec2⊕Eu(fj(·))

∥∥∥ ≤ λn

we obtain for all n ≥ n0 that∥∥∥Dfn∣∣Ec1(z)

∥∥∥ ≤ n−1∏j=0

∥∥∥Df j∣∣Ec1(z)

∥∥∥ ≤ (λλ−1)n, for all z ∈ [x, y]

and ∥∥∥Dfn∣∣Ec1(w)

∥∥∥ ≤ (λλ−1(1 + ζ))n

= λn, for all w ∈ B∞2k0ε(x).

In this way, since yc = γc2loc(x) ∩ γc1loc(y) and γc1loc(y) ⊂ B∞2k0ε(x) is tangent

to Ec1 we have

d(fn(y), fn(yc)) ≤ λnd(y, yc) for all n ≥ n0.



Now we are ready to prove the theorem.

Proof of Theorem C. Fix some small ε < ε1/6k20.

Suppose that dim(Ec) = 1. Given x ∈ M , due to Proposition 2, there

exists some central curve γcloc(x) such that B∞ε (x) ⊂ γcloc(x). We are fixing

this central curve.

As f j(B∞ε (x)) ⊂ γcloc(fj(x)) are curves of length bounded by 2ε, given

α > 0, fixed n ∈ N, on each step we can partition f j(B∞ε (x)) into b2ε/αcintervals Ik(j) of length α. Since f is a local diffeomorphism, if we take

the intersection⋂

0≤j≤n f−j(Ik(j)) and colect one point on each element we

produce a (n, α)-generator for B∞ε (x) of cardinality⌊

2εα

⌋n.

Thus, for all α > 0 we have

r(B∞ε (x), α) = limn→∞

1

nlog rn(B∞ε (x), α) ≤ lim

n→∞

1

nlog

2ε

αn = 0.

Therefore htop(B∞ε (x)) = 0 for all x ∈M if dim(Ec) = 1.

Now, assume dim(Ec) = 2. Given x ∈ M, if there are only one γc1loc(x)

tangent to Ec1 and only one γc2loc(x) tangent to Ec2 such that B∞ε (x) ⊂ γc1loc(x)

or B∞ε (x) ⊂ γc2loc(x) then with the same argument above we already have that

htop(B∞ε (x)) = 0. Hence we can suppose that fixed some γc2loc(x) there exist

y ∈ B∞ε (x) with γc1loc(y) ∩ γc2loc(x) = yc 6= y. In particular, B∞k0ε(x) ∩ γc2loc(x) 6={x}.

Given α > 0 fix n0 = n0(x) ∈ N (possibly n0 more than of the Lemma 4)

such that 3λnε < α for all n > n0.

From the continuity and transversality of Ec1 and Ec2 , let ξ ≤ α/3 be a

constant satisfying

d(z, w) ≤ ξ =⇒ d(zc, wc) ≤ α/2

where zc = γc1loc(z) ∩ γc2loc(wc) for any wc ∈ γc1loc(w).

For each n > n0 let Fn be a (n, ξ)-generator for B∞k0ε(x) ∩ γc,2loc(x). Since

this intersection is one-dimensional, Fn can be take with cardinality less than

b2k0ε/ξcn. Call the points of Fn by zi.

Now, for each point zi fixe some γc1loc(zi) and take Fn0(i) be a (n0, α/2)-

generator for B∞k0ε(x) ∩ γc1loc(zi).

Define Fn :=⋃i Fn0(i). Note that this set has cardinality less than

K b2ε/αcn where K is a constant which depends only on n0 and α.



We will show that Fn is a (n, α)-generator for B∞ε (x).

Given y ∈ B∞ε (x) let yc ∈ γc1loc(y)∩ γc2loc(x). As yc ∈ B∞k0ε(x)∩ γc2loc(x) there

exist zi ∈ Fn such that

d(f j(yc), fj(zi)) ≤ ξ for all 0 ≤ j ≤ n.

Let z = γc2loc(y) ∩ γc1loc(zi). Note that d(f j(z), f j(y)) ≤ α/2 for 0 ≤ j ≤ n

because f j(y) ∈ γc1loc(f j(yc)) and d(f j(yc), fj(zi)) ≤ ξ.

Since Fn0(i) is a (n0, α/2)-generator for B∞k0ε(x)∩γc1loc(zi) there exist zik ∈Fn0(i) such that d(f j(z), f j(zik)) ≤ α/2 for 0 ≤ j ≤ n0.

We claim that d(f j(y), f j(zik)) ≤ α for all 0 ≤ j ≤ n.

In fact, for 0 ≤ j ≤ n0 we have

d(f j(y), f j(zik)) ≤ d(f j(y), f j(z)) + d(f j(z), f j(zik)) ≤α

2+α

2= α

and for n0 < j ≤ n, by Lemma 4, we have

d(f j(y), f j(zik)) ≤ d(f j(y), f j(yc)) + d(f j(yc), fj(zi)) + d(f j(zi), f

j(zik))

≤ λjd(y, yc) + d(f j(yc), fj(zi)) + λjd(zi, zik)

≤ λjε+ ξ + λjε ≤ α

Therefore, our assertion is true.

In this way, r(B∞ε (x), α) = 0 because for each n > n0 we have find a

(n, α)-generator Fn for B∞ε (x) with cardinality K b2ε/αcn.

As α is arbitrary we conclude htop(B∞ε (x)) = 0. Since this is valid for

every x ∈M we finish the proof of Theorem C.


CHAPTER 3

Uniqueness of Equilibrium States

In this chapter, we are interesting on the uniqueness of equilibrium states for

local diffeomorphisms. In our setting it is enough to require hyperbolicity

condition on the potentials.

Let f : M →M be a C1 local diffeomorphism on a compact manifold M .

Given a positive constant σ we denote by Σσ = Σ(σ) the set whose points

satisfy:

Σσ :=

{x ∈M ; lim sup

n→+∞

1

n

n−1∑i=0


∥∥ ≤ log σ

}

We say that a real continuous function φ : M → R is a hyperbolic potential

if there exists some σ ∈ (0, 1) such that

Pf (φ,Σcσ) < Pf (φ,Σσ) = Pf (φ).

In this context, we establish

Theorem A. Let f : M → M be a transitive C1 local diffeomorphism in a

compact manifold M and let φ : M → R be a hyperbolic Holder continuous

potential. Then, there exists a unique equilibrium state µφ for (f, φ).

Our strategy to prove this theorem will be to construct an expanding

conformal measure ν associated to (f, φ). Next, we will show that there exists

19


only one f -invariant ergodic measure µφ which is absolutely continuous with

respect to ν. Finally, we will prove that this measure µφ is in fact the only

equilibrium state for (f, φ).

3.1 Conformal Measures

Let C(M) be the complete metric space of the real continuous function

ψ : M → R equipped with uniform convergence norm. The Ruelle-Perron-

Frobenius transfer operator Lφ : C(M) → C(M) associated to dynamics

f : M →M and a real continuous function φ : M → R is the linear operator

defined on C(M) by

Lφ(ψ)(x) =∑

y∈f−1(x)

eφ(y)ψ(y)

Since f is a local homeomorphism defined on a compact metric space and

φ is a continuous function follows that Lφ is well defined. Moreover, Lφ is a

continuous operator:∣∣∣∣Lφ(ψ)(x)

∣∣∣∣ =

∣∣∣∣ ∑y∈f−1(x)

eφ(y)ψ(y)

∣∣∣∣ ≤ e‖φ‖∞deg(f)|ψ(x)|, for all x in M.

And

‖Lφ‖ ≤ deg(f)e‖φ‖∞

Note that, proceeding by induction we have for all n ≥ 1 :

Lnφ(ψ)(x) =∑

y∈f−n(x)

eSnφ(y)ψ(y)

where Snφ denotes the Birkhoff sum Snφ =n−1∑k=0

φ ◦ fk.

Let (C(M))∗ be the dual space of C(M). Since M is compact we can

identify (C(M))∗ with M(M) the space of finite signed measure defined on

M , i.e., (C(M))∗ ' M(M). Therefore, the dual operator L∗φ of Lφ acts on

(C(X))∗ as follow:

L∗φ : (C(M))∗ 'M(M) −→ (C(M))∗ 'M(M)

µ 7−→ L∗φ(µ) : C(M) −→ R

ψ 7−→ L∗φ(µ)(ψ) =

∫M

Lφ(ψ)dµ



Let Φ :M(M)←↩ be the operator defined on M(M) by

Φ(µ) =L∗φ(µ)∫

MLφ(1)dµ

Note that Φ is a continuous operator onM(M) because L∗φ is continuous

and

Lφ(1) =∑

y∈f−1(x)

eφ(y) ≥ deg(f)e−||φ||∞ = c > 0

implies ∫M

Lφ(1)dµ ≥ cµ(M) > 0, ∀µ ∈M(M).

In particular, Φ gives invariant the convex compact spaceM1(M) of the

probabilities measures. So, we can apply the Schauder-Tychonov theorem

and conclude that there exists a fixed point ν ∈M1(M) for Φ:

Theorem 3 (Schauder-Tychonov). Let K be a compact convex subset of a

locally convex topological vectorial space and let Φ : K → K be a continuous

aplication. Then, Φ has some fixed point.

This fixed point ν is an eigenmeasure for L∗φ :

Φ(ν) = ν =⇒ L∗φ(ν) = λν where λ =

∫M

Lφ(1)dν > 0

Next we derive some importants properties of this eigenmeasure.

The Jacobian of a measure µ with respect to f is a measurable function

Jµf satisfying

µ(f(A)) =

∫A

Jµfdµ

for any measurable set A such that f |A is injective.

In general, a jacobian may fail to exists, in the meantime it is a standard

result that a λ-eigenmeasure has jacobian and it is equal to λe−φ.

Lemma 5. Let ν be a Borel probability measure such that L∗φ(ν) = λν for

some λ > 0. Then, the Jacobian of ν with respect to f is a continuous function

given by Jνf = λe−φ. Moreover, supp(ν) = M if f is transitive.

Proof. Let A be a measurable set such that f |A is injective. Pick a sequence

{ψn} ∈ C(M) such that sup |ψn| ≤ 2 for all n ∈ N and ψn → χA for ν-qtp.

Then,

Lφ(e−φψn)(x) =∑

y∈f−1(x)

eφ(y)e−φ(y)ψn(y) =∑

y∈f−1(x)

ψn(y).



Applying the Dominated Convergence theorem we have∫M

λe−φψndν =

∫M

e−φψnd(L∗φν) =

∫M

Lφ(e−φψn)dν −→ ν(f(A))

Since the first member converges to∫Aλe−φdν, we conclude that

ν(f(A)) =

∫A

λe−φdν.

Now suppose that f is transitive. Given U ⊂ M an open set, by transi-

tivity property we have M =⋃s∈N f

s(U).

Since for each s, f s is also a local homeomorphism we can decompose U

into subsets Vi(s) ⊂ U such that f s|Vi(s) is injective. Hence,

1 = ν(M) ≤∑s

ν(f s(U)) ≤∑s

∑i

∫Vi(s)

λse−Ssφ(x)dν

≤∑s

λs∑i

supx∈Vi(s)

(eSsφ(x))ν(Vi(s))

Thus, there exists some Vi(s) ⊂ U such that ν(U) ≥ ν(Vi(s)) > 0.

In this way, supp(ν) = M because ν is an open measure.

We call the eigenmeasure ν a conformal measure associated to (f, φ).

From now, we will consider hyperbolic potentials φ : M → R, that mean,

there exist some σ ∈ (0, 1) such that

Pf (φ,Σcσ) < Pf (φ,Σσ) = Pf (φ).

The above inequality allows us to explore the non-uniform expansion

property on Σ. In this direction, we will use frequently the hyperbolic times.

For remenber this property see the preliminaries.

Fix δ > 0 such that the dynamical ball B(x, n, δ) is mapped diffeomor-

phically by fn onto the ball B(x, δ) whenever n is a hyperbolic time for x.

Lemma 6. Fixed ε ≤ δ there exists Kε > 0 such that if n is a hyperbolic

time for x then

K−1ε ≤

ν(Bε(x, n))

eSnφ(y)−n log λ≤ Kε

for all y in the dynamical ball Bε(x, n).



Proof. Let n be a hyperbolic time for x. Since fn maps homeomorphically

Bε(x, n) into the ball B(fn(x), ε) and the jacobian of ν is bounded away from

zero and infinity follows that there exists a uniform constant γε depends on

the radius ε of the ball such that

γε ≤ ν(fn(Bε(x, n))) =

∫Bε(x,n)

λne−Snφ(z)dν ≤ 1

By distortion control on hyperbolic times (Lemma 2), we have

γε ≤∫Bε(x,n)

λne−Snφ(z)dν =

∫Bε(x,n)

λne−Snφ(y)

(λne−Snφ(z)

λne−Snφ(y)

)dν

≤ Ke−Snφ(y)+n log λν(Bε(x, n))

and

e−Snφ(y)+n log λν(Bε(x, n)) ≤ K

∫Bε(x,n)

λne−Snφ(y)

(λne−Snφ(z)

λne−Snφ(y)

)dν

Thus,

γεK−1 ≤ ν(Bε(x, n))

eSnφ(y)−n log λ≤ K

for all y ∈ Bε(x, n).

In the following proposition, it will be explicit the eigenvalue λ. For this,

we will explore the fact of pressure of φ is located on a non-uniformly expan-

ding set.

Proposition 3. Let ν be a λ-eigenmeasure associated to L∗φ. Assume that

φ : M → R is a hyperbolic Holder continuous potential. Then, ν is an

expanding measure and the eigenvalue λ associated to ν is log λ = Pf (φ).

Proof. To see the equality Pf (φ) = log λ, let α be a cover of M with dia-

menter less than δ. Given n ∈ N denote by U a subcover of αn. As ν is an

eigenmeasure we have



λn = λnν(M) =

∫M

Lnφ(1)dν ≤∑U⊂U

∫U

eSnφ(x)dν

≤∑U⊂U

eSnφ(U)ν(U)

≤∑U⊂U

eSnφ(U)

Taking the infimum of all subcover U of αn we obtain the inequality

log λ ≤ 1

nlog inf

U⊂αn

{∑U⊂U

eSnφ(U)}

n→+∞−→ Pf (φ, α) ≤ Pf (φ)

The converse inequality will be consequence of the relative pressure on

Σσ to be equal to the pressure of the system.

Let Σj be the set of points in Σσ that have j ≥ 1 as hyperbolic time.

Note that, fixed ε < δ, we can cover Σj by dynamical balls Bε(w, j) centered

on points w ∈ Σj.

Since f is a transitive local homeomorphism, given y ∈ Σ, there exists

some s = s(ε), such that only one pre-image z = f−s(y) is in the open ball

B2ε(fj(w)) . Applying the contraction property on hyperbolic times we have

f−j(z) ∈ Bε(w, j). Thus, we can cover each Σj by disjoint dynamical balls{Bε(f

−j(z), j); z = f−s(y)}

for every j ∈ N.

In this way, fixed y ∈ Σ we have a covering:

Σ ⊂⋃j≥0

Σj ⊂⋃j≥0

⋃z=f−s(y)

Bε(f−j(z), j)

Fix some γ > log λ. Given any n ≥ 1 let

Un :=⋃j≥n

⋃z=f−s(y)

Bε(f−j(z), j)



Using the previous estimate of the ν-measure of dynamical balls, we have∑j≥n

e−γj+Sjφ(Bε(f−j(z),j)) ≤∑j≥n

Ke−(γ−log λ)j{ ∑z=f−s(y)

ν(Bε(f−j(z), j))

}≤ K

∑j≥n

e−(γ−log λ)j

≤ Ke−(γ−log λ)n

Taking the limite on n we obtain

mf (φ,Λ, ε, γ) = limn→+∞

mf (φ,Λ, ε, n, γ) = 0

As this limite is true for every ε > 0 and γ > log λ we have

Pf (φ,Σσ) ≤ log λ

Since by hypothesis, Pf (φ,Σcσ) < Pf (φ,Σσ) = Pf (φ) we conclude

Pf (φ) = Pf (φ,Σσ) ≤ log λ

Remains to check that ν is expanding. Fixed ε small, let FN be the

collection of dynamical balls

FN = {Bδ(x, n) / x ∈M and n ≥ N}

By definition of pressure and the conformality of ν, for every finite or count-

able family U of FN which cover Σcσ holds:

ν(Σcσ) ≤ ν(U) ≤

∑Bε(x,n)∈U

λ−neSnφ(Bε(x,n))

=∑

Bε(x,n)∈U

e−(log λ)n+Snφ(Bε(x,n))

Taking the infimum over FN we have

ν(Σcσ) ≤ mf (φ,Σ

cσ, N, log λ).

Since Pf (φ,Σcσ) < Pf (φ) = Pf (φ,Σ) = log λ, as N goes to infinity we

obtain:

ν(Σcσ) ≤ lim

N→+∞mf (φ,Σ

cσ, N, log λ) = mf (φ,Σ

cσ, log λ) = 0.



We complete this section with a general result which is a consequence

of non-uniform expansion on Σ. This proposition will allow us to show the

uniqueness of equilibrium measure.

Proposition 4. Let f : M →M be a transitive C1 local diffeomorphism and

φ : M → R be a hyperbolic Holder continuous potential. Then, there exists

only one f -invariant ergodic measure µφ absolutely continuous with respect

to the conformal measure ν associated to (f, φ).

Proof. Let (µn)n be the sequence of averages of the positive iterates of the ν

measure restricted to Σ :

µn :=1

n

n−1∑j=0

f j∗ (ν|Σ)

Denoting by Σj the set of points in Σ that have j ≥ 1 as a hyperbolic time,

we define

ηn :=1

n

n−1∑j=0

f j∗ (ν|Σj)

Applying Lemma 1 there are some θ > 0 and n0 ∈ N such that for all n ≥ n0

holds:

ηn(M) ≥ 1

n

n−1∑j=0

ν(Σj) ≥1

n

n−1∑j=0

ν(Σ ∩ Σj) ≥ θν(Σ).

Since the jacobian Jνf of the measure ν is a positive function bounded

from below follows that each ηn is absolutely continuous with respect to ν.

Take a subsequence nk →∞ such that µnk and ηnk converge in the weak*

topology to measures µ and η respectively. Therefore µ is an f -invariant

measure which has a component η absolutely continuous with respect to ν.

Hence, if we decompose µ = µac + µs with µac absolutely continuous with

respect to ν and µs singular to ν we have µac(Σ) ≥ η(Σ) > 0 and µac is an

f -invariant measure absolutely continuous with respect to ν.

To see the uniqueness, we will use the well known fact that every f -

invariant set A with positive ν-measure ν(A) > 0 is contained ν-qtp in a

topological disk ∆, i.e., there exists some disk ∆ such that ν(∆\A) = 0. For

a reference of this result consult [1].

By contradiction, suppose that (f, φ) has two distinct µ1 and µ2 f -

invariant ergodic measures absolutely continuous with respect to ν. Denote



by B(µ1) and B(µ2) the basins of attraction of the measures µ1 and µ2, re-

spc. Since these sets are positively invariant, there are topological disks ∆1

and ∆2 such that ν(∆i \B(µi)) = 0 for i = 1, 2. Therefore the transitivity of

f and the invariance of B(µ1) and B(µ2) imply that ν(B(µ1) ∩ B(µ2)) > 0.

Since distinct ergodic measures have disjoint basins we have a contradiction.

Note that, since ν is expanding then µφ is expanding too.

3.2 Proof of Theorem A

The purpose of this section is to prove the Theorem A. Until now, we cons-

truct a conformal measure ν associated to (f, φ) and we show the existence

of an unique f -invariant ergodic measure µφ absolutely continuous to ν. Our

next step will be to show that µφ is the only equilibrium state for (f, φ).

Proposition 5. The measure µφ is an equilibrium state for (f, φ).

Proof. Firstly, note that, since µφ is a f -invariant measure absolutely con-

tinuous with respect to ν, the density h := dµφ/dν is an L1(ν) function and

it satisfies:∫gLφ(h)dν =

∫Lφ(g ◦ f · h)dν =

∫g ◦ f · hdL∗φ(ν) = λ

∫g ◦ f · h dν

= λ

∫g ◦ f dµ = λ

∫g dµ =

∫g · λhdν

for every continuous function g. Thus, Lφ(h) = λh in ν almost every where.

We claim that there exists some positive constant K such that h(y) ≥K > 0 for almost every where y ∈ Σ.

In fact, by Proposition 4, h = dµφ/dν ≥ dη/dν where η is an acumulation

point of the sequence

ηn :=1

n

n−1∑j=0

f j∗ (ν|Σj)

Moreover, fixed some ε < δ, given y ∈ Σ, by the density of its preimage,

there exists some s depending only ε such that we can cover each Σj by



dynamical balls {Bε(f−j(z), j) / z ∈ f−s(y)}. Thus, for every n > 0 we have

∑z∈f−s(y)

dηndν

(z) =∑

z∈f−s(y)

1

n

n−1∑j=0

df∗(ν|Σj)dν

(z)

=∑

z∈f−s(y)

1

n

n−1∑j=0

{ ∑w∈f−j(z)

λ−jeSjφ(w)}

≥∑

z∈f−s(y)

1

n

n−1∑j=0

{ ∑w∈f−j(z)

K−1ν(Bε(w, j))}

≥ K−1 1

n

n−1∑j=0

ν(Σj)

≥ K−1θ

On the other hand, as Lφ(h) = λh we conclude our claim

h(y) = λ−s∑

z∈f−s(y)

eSsφ(z)h(z) ≥ K2λ−ses inf φ > 0.

Now, given a borelean A ⊂M , by definition of jacobian we have:∫A

Jµφf dµφ = µ(f(A)) =

∫f(A)

h(y) dν(y)

=

∫A

h ◦ f(x) · Jνf(x) dν(x)

=

∫A

h ◦ f · Jνfh

(x) d(hν)(x)

=

∫A

λe−φ · h ◦ fh

(x) dµ(x)

Hence, for µφ every point holds:

Jµφf (x) =λe−φ · h ◦ f

h(x)

On the other hand, since µφ is an expanding measure, by Roklhin’s for-

mula we have

hµφ(f) =

∫log Jµφf dµφ.



In particular,

logh ◦ fh

= log Jµφf − log Jνf

is an L1(µφ) function and by Birkhoff ergodic theorem we have∫log

h ◦ fh

dµφ = 0

In this way, we conclude that µφ is an equilibrium measure for (f, φ) :

hµφ(f) +

∫φ dµφ =

∫log Jµφf dµφ +

∫φ dµφ

=

∫log λe−φdµφ +

∫log h ◦ f

log hdµφ +

∫φ dµφ

= log λ−∫φ dµφ +

∫φ dµφ

= Pf (φ)

To finish the proof of the Theorem A remains to check that µφ is the

unique equilibrium state.

Let η be an ergodic equilibrium state for (f, φ), that means,

log λ = Pf (φ) = hη(f) +

∫φ dη.

Let τ be a partition for Σ with diamenter less than δ. Hence, f is injective

on each atom of τ and τ is a generating partition with respect to any mesure

µ such that µ(Σ) = 1 because every point in Σ has infinitely many hyperbolic

times.

Define the partition ξ by ξ :=∨n≥0 f

nτ . Observe that f−1ξ is finer than

ξ and∨n≥0 f

−nξ is also a generating partition. We will compare µφ and η

with respect to this partition ξ.

Denote by (ηx)x and by (µx)x the Rokhlin’s disintegration of the measures

η and µφ on the partition ξ, respect.

Since µφ is equal to hν its decomposition (µx)x over ξ is given by:

µx(A) =1

µ(ξ(x))

∫A∩ξ(x)

hx(y) dνx(y)



Thus, fixed n > 0, since f−1ξ is finer than ξ we have

µx(f−nξ(x)) =

1

µ(ξ(x))

∫f−nξ(x)

hx(f−n(y)) dνx(f

−n(y))

Using change of coordinates, we compute

µx(f−nξ(x)) =

1

µ(ξ(x))

∫f−nξ(x)

hx(f−n(y)) dνx(f

−n(y))

=1

µ(ξ(x))

∫ξ(x)

hx(f−n(y))λ−neSnφ(x) dνx(f

−n(y))

=1

µ(ξ(x))

∫fnξ(x)

hx(f−n(y))λ−neSnφ(y)

λne−Snφ(x)dνfn(x)(y)

=1

µ(ξ(x))

∫fnξ(x)

hfn(x)(y)

λne−Snφ(x)dνfn(x)(y)

=µφ(fnξ(x))

µφ(ξ(x))Jνfn(x)

on the fourth equality we use the fact that Lφ(h) = λh in almost every where.

Note that, log µx(f−nξ(x)) is a non-positive function and therefore it

has an η-integrable positive part. Hence, applying again Birkhoff ergodic

theorem, we see

−∫

log µx(f−n(ξ)(x)) dη(x) =

∫log Jνf

n(x) dη(x)

=

∫log λne−Snφ(x) dη(x)

= n(log λ−∫φ dη)

= nhη(f)

On the other hand, since ξ is an generating partition follows that

nhη(f) = nhη(f, ξ) = Hη(f−nξ|ξ) = −

∫log ηx(f

−n(ξ)(x)) dη(x)

Therefore, by the concavity of the logarithm, we obtain the equality



0 = −∫

logµx(f

−n(ξ)(x))

ηx(f−n(ξ)(x))dη(x) =

∫log

dµφdη

dη

≤ log

∫dµφdη

dη = 0

Since this equality can hold only if the above derivative is constantly

equal to 1 we conclude that η = µφ on the σ-algebra given by f−n(ξ). As

f−n(ξ) goes to the total σ-algebra when n → ∞, we prove that η = µφ and

this finishes the proof of Theorem A.

We observe that the Theorem A applies also to local homeomorphisms

f : M → M of compact metric space M and Holder continuous potentials

φ : M → R hyperbolic in the sense of the pressure is located on a non-

uniformly expanding region Σ ⊂M , that means,

Pf (φ,Σc) < Pf (φ,Σ) = Pf (φ)

where

Σ = {x ∈M / lim supn→+∞

1

n

n−1∑j=0

log σ(f j(x)) ≤ log σ < 0}

and σ is a function that to each point x ∈M associate the Lipschitz constant

of the local inverse branch fx : Vx → f(Vx), i.e.,

d(f−1x (y), f−1

x (z)) ≤ σ(x)d(y, z)

Indeed, all intermediates results presented here, such as, the existence of

conformal measures, distortion control, uniqueness of absolutely continuous

invariant ergodic measures, remain true in this setting.


CHAPTER 4

Partially Hyperbolic Skew-Products

We close this work giving suficient conditions to uniqueness of equilibrium

states for skew-products. Under the hypotheses of non-uniform expansion in

some region of the base and uniform contraction on the fiber we obtain the

uniqueness of equilibrium measures. Let us start setting the dynamics.

Let M be a compact manifold and N be a complete metric space. Let F

be the skew-product:

F : M ×N →M ×N , F (x, y) = (f(x), gx(y))

with f : M → M a C1 transitive local diffeomorphism and g : M ×N → N

a fiber contraction that varies β-Holder continuously on M , i.e., there are

constants C1 > 0, λ ∈ (0, 1) and β ≤ 1 such that

dN(gx(y), gx(z)) ≤ λdN(y, z) for all y, z ∈ N and x ∈M

and

dN(gx(y), gz(y) ≤ C1dM(x, z)β for all x, z ∈M and y ∈ N

Due to uniform contraction of g, the dynamics F has an attractor Λ ⊂M×N .

Let γ > 0 be a constant such that γ ≤ ‖Df(x)‖ for all x ∈M. Fix δ < 1

some positive constant such that δβγ−1 < 1. If there exists σ ∈ (0, 1) such

that λγ−1 ≤ λδ−β ≤ σ < 1 for all x ∈ M we call Λ a partially hyperbolic

attractor for F .

32


Remark. If N is a complete n-dimensional manifold, f is partially hyper-

bolic of type Ec⊕Eu and g is differentiable with ‖Dyg‖ ≤ λ for all y ∈ Λ then

Λ is a partially hyperbolic attractor in the usual sense. But in our work, it

is not necessary N to be finite dimensional and f to be partially hyperbolic.

From now on Λ will be a partially hyperbolic attractor for the skew-

product F (x, y) = (f(x), gx(y)). The main goal of this chapter is to prove

that when we join our hyperbolicity hypothesis of φ on the bases M , that

means, PF (φ,Σc ×N) < PF (φ,Σ×N) = PF (φ), we obtain

Theorem B. Let Λ be a partially hyperbolic attractor for the skew-product

F : M × N → M × N , F (x, y) = (f(x), gx(y)) and let φ : M × N → Rbe a Holder continuous potential hyperbolic on M . Then, there is only one

equilibrium state µφ associated to (F, φ).

The strategy to prove Theorem B is to extend the skew-product endomor-

phism F to the skew-product homeomorphism F via the natural extension

f of f . Due to the uniform contraction of g on the fibers, we will show that

F has an attractor defined by a graph of a Holder function. Hence we can

understand the dynamics of F from the dynamics of f . Next, for each Holder

continuous potential φ : M ×N → R for F , hyperbolic on the bases, we will

construct a hyperbolic Holder continuous potential ψ : M → R for f . Thus,

the uniqueness of the equilibrium states for (F, φ) will be consequence of the

uniqueness for (f , ψ), given by Theorem A.

Let f be the natural extension of f ,

f : M → M, f(x) = f(..., x2, x1, x0) = (..., x2, x1, x0, f(x0))

Since δ < 1 is fixed, we will consider M provided with the metric

dM(x, y) :=∑j≥0

δjdM(xj, yj).

Define F : M ×N → M ×N by F (x, y) = (f(x), gπ(x)(y)).

In order to avoid overloading the notation, we will denote gπ(x)(y) by

g(x, y). However, it is very important to note that this is just a notation.

Clearly, g depends only on x and not of his past, that means, if π(x) = π(z)



then g(x, y) = g(z, y) for all y ∈ N. Moreover, we will denote by gn the

iterates of g :

gn(x, y) := g(fn(x), gn−1(x, y)) with g1(x, y) = g(x, y) for all (x, y) ∈ M×N.

Note that, f is a homeomorphism dominated by g, because for all n ∈ N

dM(f−n(x), f−n(z)) ≤ δ−ndM(x, z) and λδ−1 ≤ σ < 1

Furthermore, g is fiber contraction which varies β-Holdercontinuously on M :

dN(g(x, y), g(z, y)) ≤ C1dM(x, z)β ≤ C1dM(x, z)β

for all x, z ∈ M and y ∈ N.

The following proposition due to Hirsch, M., C. Pugh and M. Shub [11]

ensures that F has an attractor Λ given by a graph of a Holder continuous

function. Let us take a sketch of the proof for completeness of the work.

Proposition 6. There exists a Holder function ξ : M → N such that the

graph of ξ is F -invariant and attracting for all (x, y) ∈ M ×N .

Proof. We would like to show the existence of a Holder function ξ : M → N

such that denoting by Λ := graph(ξ) ={

(x, ξ(x)) : x ∈ M}

we have

d(F n(x, y), F n(x, ξ(x))

)−→ 0 as n→ +∞ for all (x, y) ∈ M ×N.

If such F -invariant graph exists, it has satisfy:

F (x, ξ(x)) = (f(x), g(x, ξ(x))) = (f(x), ξ(f(x)))

Thus, ξ(x) = g(f−1(x), ξ(f−1(x)) for all x ∈ M.

In this way, we are going to search a fixed point of the graph transform:

Γ : ψ ∈ C0(M,N) 7−→ g ◦ (Id, ψ) ◦ f−1 ∈ C0(M,N)

where C0(M,N) denote the complete metric space of the continuous function

ψ : M → N equipped with the usual uniform convergence norm.

Note that, Γ is well defined because given ψ ∈ C0(M,N) we have Γ(ψ) is

also in C0(M,N) since Γ(ψ) is a composition of continuous function.

Let us display the function ξ : M → N in two steps: first, its existence,

second, its regularity.



Given ψ, ψ′ ∈ C0(M,N),

d(Γ(ψ)(x),Γ(ψ

′)(x)

)= d

(g(f−1(x), ψ(f−1(x))

), g(f−1(x), ψ

′(f−1(x))

))≤ λd

(ψ(f−1(x)), ψ

′(f−1(x))

)≤ λd

(ψ, ψ

′)Taking the supremum over all x ∈ M we have d

(Γ(ψ),Γ(ψ

′))≤ λd

(ψ, ψ

′).

Since λ < 1 follows from Contraction Mapping Theorem that Γ has a

unique fixed point ξ ∈ C0(M,N). By construction,

ξ(f(x)) = Γ(ξ)(f(x)) = g(f−1(f(x)), ξ(f−1(f(x)))) = g(x, ξ(x))

Therefore, the graph of ξ is F -invariant.

By induction, we have ξ(fn(x)) = gn(x, ξ(x)) and so

d(F n(x, y), F n(x, ξ(x))

)= d

((fn(x), gn(x, y)), (fn(x), gn(x, ξ(x)))

)≤ λnd

(y, ξ(x)

)−→ 0 as n→ +∞

Thus, the graph of ξ is attracting under F . Remains to show that ξ is a

Holder function.

Denote by Cυk (M,N) ⊂ C0(M,N) the space of Holder functions with

constant k and exponent υ, namely,

Cυk (M,N) :={ψ : M → N : d(ψ(x), ψ(z)) ≤ kd(x, z)υ for all x, z ∈ M

}The space Cυk (M,N) is closed in C0(M,N) on the topology of uniform

convergence for any constant k and any exponent υ.

Let k0 ≥ C1δ−β(1− λδ−β)−1.

We claim that Γ(Cβk0(M,N)) ⊂ Cβk0(M,N).

Indeed, given ψ ∈ Cβk0(M,N) we have:

d(Γ(ψ)(x),Γ(ψ)(z)

)= d

(g(f−1(x), ψ(f−1(x))

), g(f−1(z), ψ(f−1(z))

))≤ d

(g(f−1(x), ψ(f−1(x))

), g(f−1(x), ψ(f−1(z))

))+ d

(g(f−1(x), ψ(f−1(z))

), g(f−1(z), ψ(f−1(z))

)).

Since,

d(g(f−1(x), ψ(f−1(x))

), g(f−1(x), ψ(f−1(z))

))≤ k0λd(f−1(x), f−1(z))β

≤ k0λδ−βd(x, z)β.



and

d(g(f−1(x), ψ(f−1(z))

), g(f−1(z), ψ(f−1(z))

))≤ C1d(f−1(x), f−1(z))β

≤ C1δ−βd(x, z)β

Follows that,

d(Γ(ψ)(x),Γ(ψ)(z)

)≤

(k0λδ

−β + C1δ−β)d(x, z)β

≤ k0d(x, z)β

Hence, Γ(Cβk0(M,N)) ⊂ Cβk0(M,N) as we claim.

Finally, fix some point y0 ∈ N and let ψ0 : M → N be the constant

function ψ0(x) = y0. Define the sequence ψn = Γn(ψ0). Then ψn ∈ Cβk0(M,N)

for all n ∈ N and ψn → ξ because

d(ψn, ξ

)= d(Γn(ψ0),Γn(ξ)

)≤ λnd

(ψ0, ξ

)−→ 0 as n→ +∞

Since Cβk0(M,N) is closed follows that ξ is a Holder function.

This proposition allows us to reduce the dynamics F to the dynamics f ,

namely, given a potential φ : M × N → R, since graph(ξ) is an attracting

invariant set we have PF (φ) = PF(φ∣∣graph(ξ)

)and φ

∣∣graph(ξ)

is a potential

for f , so there exists a correspondence between equilibrium measures. In

particular, (F , φ) will have only one equilibrium state due to the uniqueness

for (f , φ∣∣graph(ξ)

).

Now we are ready to prove the theorem.

Proof of Theorem B. Let F : M × N → M × N be a partially hyperbolic

skew-product F (x, y) = (f(x), gx(y) ) and φ : M × N → R be a α-Holder

continuous potential, hyperbolic on M .

Define the potential φ : M × N → R by φ(x, y) = φ ( π(x), y) where

π : M → M is the natural projection. Notice that φ is Holder continuous

on M ×N :

dR(φ(x, y), φ(z, w)) = dR(φ(π(x), y), φ(π(z), w))

≤ k dM×N((π(x), y), (π(z), w))α

≤ k dM×N((x, y), (z, w))α

for all (x, y), (z, w) ∈ M ×N.



Claim. There exists a unique equilibrium state µφ associated to (F , φ).

Proof of the Claim. Firstly we observe that since f is a local diffeomorphism

then the natural extension f−1 is locally Lipschitz continuous, i.e., given

x ∈ M there exists a neighborhood Vx such that for every y, z ∈ f(Vx) we

have

dM(f−1(y), f−1(z)) ≤ σ(x) dM(y, z)

where σ(x) = ‖Df−1‖ ◦ π(x).

In particular, if Σ is a non-uniformly expanding region on M then for

every x ∈ π−1(Σ) holds

lim supn→+∞

1

n

n−1∑j=0

log σ(f j(x)) ≤ log σ < 0

Hence, f is a non-uniformly expanding on Σ := π−1(Σ).

Furthermore, since Rec(F ) = graph(ξ) and ξ : M → N is a Holder

continuous function follows that φ induces a Holder continuous potential ψ

for f defined by

ψ : M → R, ψ(x) := φ(x, ξ(x)) with Pf (ψ) = PF (φ∣∣graph(ξ)

) = PF (φ)

Note also fixed ε > 0 and n0 ∈ N, if Fn0 is the collection of dynamical

balls

Fn0 = {Bε(x, n) / x ∈ M and n ≥ n0}

that cover Σc ⊂ M then the collection Fn0 of dynamical balls

Fn0 = (π, ξ)(Fn0) = {(π, ξ)(Bε(x, n))/π(x) ∈M, ξ(x) ∈ N andn ≥ n0}

cover Σc × ξ(Σc) ⊂M ×N and for every α ∈ R holds

infBε(x,n)

∑n≥n0

e−αn+Snψ(Bε(x,n) ≤ infBε((x,ξ(x)),n)

∑n≥n0

e−αn+Snφ((π,ξ)(Bε(x,n)))

Thus for every ε holds

Pf (ψ, Σc, ε) ≤ PF (φ,Σc ×N, ε)

So,

Pf (ψ, Σc) ≤ PF (φ,Σc ×N) < PF (φ,Σ×N)

= PF (φ)

≤ PF (φ) = Pf (ψ)



In this way, ψ is a hyperbolic potential for f . Hence, by Theorem A, there

exists a unique equilibrium state µψ for (f , ψ).

Let π : M ×N → M be the canonical projection. Since g is an uniform

contraction in the fiber π−1(x) follows that htop(π−1(x)) = 0 for all x ∈ M.

Thus, we can apply the Ledrappier-Walter’s formula (see preliminaries) to

conclude that the measure µφ such that π∗µφ = µψ is an equilibrium state

for (F , φ):

PF (φ) = Pf (ψ) = hπ∗µφ(f) +

∫ψdµπ∗µφ

= hµφ(F )− (supx∈M

htop(π−1(x))) +

∫φ∣∣graph(ξ)

dµφ

= hµφ(F ) +

∫φdµφ

Moreover, by the uniqueness of µψ, if there exists two ergodic equilibrium

states µ1φ

and µ2φ

for (F , φ), as we saw above, we must have π∗µ1φ

= µψ = π∗µ2φ.

Denote by Bµ1φ(F ) and Bµ2

φ(F ) the basins of attraction of µ1

φand µ2

φ

respectively. Since g is a fiber contraction and µ1φ, µ2

φare ergodic measures

follows that

Bµ1φ(F ) = A1 ×N, Bµ2

φ(F ) = A2 ×N with A1 ∩ A2 = ∅

From the ergodicity of the measure µψ and the f -invariance of the sets

π(Bµφ(F )) and π

(Bµ2

φ(F ))

we conclude that

µψ(π(Bµ1

φ(F ))

= µψ(π(Bµ2

φ(F )))

= 1

Thus,

π(Bµ1

φ(F ))∩ π(Bµ2

φ(F ))6= ∅

imply, A1 = A2. Hence µ1φ

= µ2φ

and we prove the Claim.

Let µφ =(π, id

)∗µφ be the projection of µφ by

(π, id

). In this way, µφ is

an equilibrium state for (F, φ) :

PF (φ) ≥ PF (φ) ≥ hµφ(F ) +

∫φdµφ

≥ hµφ(F )− (supx∈M

htop(π−1(x))) +

∫φdµφ

= PF (φ)



On equality, we used the fact that htop(π−1(x)) = 0 for all x ∈M because f

is the natural extension of f.

Finally, suppose η is any equilibrium state of (F, φ). Let µ ∈MF (M×N)

be an F−invariant measure such that(π, id

)∗µ = η. Then hµ(F ) ≥ hη(F )

and so

hµ(F ) +

∫φ dµ ≥ hη(F ) +

∫φ dη = PF (φ) = PF (φ).

Thus µ is an equilibrium state for (F , φ). By the Claim we must have

µ = µφ. Then η =(π, id

)∗µ =

(π, id

)∗µφ = µφ.


Bibliography

[1] J. F. Alves, C. Bonatti, and M. Viana. SRB measures for partially

hyperbolic systems whose central direction is mostly expanding. Invent.

Math., 140 (2000), 351-398.

[2] A. Arbieto, C. Matheus and K. Oliveira. Equilibrium states for random

non-uniformly expanding maps. Nonlinearity, 17 (2004), 581-593.

[3] R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomor-

phisms . Lectures Notes in Mathematics, Springer-Verlag, 470 (1975).

[4] R. Bowen. Entropy-expansive maps. Trans. A. M. S., 164 (1972).

[5] H. Bruin and G. Keller. Equilibrium states for S-unimodal maps. Ergodic

Theory and Dynam. Systems. 18 (1998), 765-789.

[6] J. Buzzi, F. Paccaut, and B. Schmitt. Conformal measures for multidi-

mensional piecewise invertible maps. Ergodic Theory and Dynam. Sys-

tems, 21 (2001) 1035-1049.

[7] J. Buzzi and O. Sarig. Uniqueness of equilibrium measures for countable

Markov shifts and multidimensional piecewise expanding maps Ergodic

Theory and Dynam. Systems, 23 (2003), 1383-1400.

[8] M. Denker and M. Urbanski. Ergodic theory of equilibrium states for

rational maps. Nonlinearity, 4 (1991), 103-134.

[9] M. Denker, G. Keller and M. Urbanski. On the uniqueness of equilibrium

states for piecewise monotone mappings. Studia Math 97 (1990), 27-36.

40


[10] L. Dıaz, T. Fisher, M. Pacıfico and J. Vieitez. Entropy-expansiveness for

partially hyperbolic diffeomorphism. Discret and Continuous Dynamical

Systems 32 (2012).

[11] M. Hirsch, C. Pugh, and M. Shub. Invariant manifolds. Lectures Notes

in Mathematics, Springer-Verlag, 583 (1977).

[12] F. Hofbauer and G. Keller. Ergodic properties of invariant measures for

piecewise monotonic transformations. Math. Z. 180 (1982), 119-140.

[13] W Krieger. On the uniqueness of the equilibrium state. Math. Systems

Theory, 8 (1974).

[14] F. Ledrappier, P. Walters. A relativised variational principle for contin-

uous transformations. J. Lond. Math. Soc., 16 (1977).

[15] R. Leplaideur, K. Oliveira and I. Rios. Invariant manifolds and equi-

librium states for non- uniformly hyperbolic horseshoes. Nonlinearity, 19

(2006), 2667-2694.

[16] G. Liao, M. Viana and J. Yang. The entropy conjecture for diffeomor-

phisms away from tangencies. preprint, arxiv: 1012.0514.

[17] K. Oliveira and M. Viana. Thermodynamical formalism for robust

classes of potentials and non-uniformly hyperbolic maps. Ergodic The-

ory and Dynam. Systems 28 (2008).

[18] Y. Pesin. Dimension theory in dynamical systems. University of Chicago

Press, Contemporary views and applications (1997).

[19] V. A. Rokhlin. Exact endomorphisms of a lebesgue space. Izv. Akad.

Nauk SSSR Ser. Mat. 25 (1961), 499-530.

[20] P. Varandas and M. Viana. Existence, uniqueness and stability of equi-

librium states for non- uniformly expanding maps. Annales de L’Institut

Henri Poincare. Analyse non Lineaire, 27 (2010), 555-593.

[21] M. Yuri. Thermodynamical formalism for countable to one Markov sys-

tems. Trans. Amer. Math. Soc., 335 (2003), 2949-2971.


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Instituto de Matem atica Pura e Aplicada - impa.br · Instituto de Matem atica Pura e Aplicada Vanessa Ribeiro Ramos EQUILIBRIUM STATES FOR HYPERBOLIC POTENTIALS Thesis presented